88 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA We have already seen (12.15.11) that (a) implies (b). If u is weakly con- tinuous, then for all y e E the linear form x\-+(u(x) \y) is weakly continuous, and therefore a fortiori continuous with respect to the strong topology, which is finer than the weak topology. Consequently (63.2), for each y e E there exists a unique point u*(y) such that (u(x)\y) = (x\u*(yj) for all xe E: in other words (11.5), u has an adjoint, and therefore (b) implies (c). Finally, if u has an adjoint, then we have Htt*GOII = sup |(*|ii*(y))|= sup \(u(x)\y)\, \\x\\£l ||*||SS1 and each of the linear forms y*-+(y\u(x)) = (u(x)\y) is continuous, hence y\~+ \\u*(y)\\ is a lower semicontinuous seminorm on E, and is therefore con- tinuous in the strong topology (12.16.3). But this is equivalent to saying that u* is (strongly) continuous, and therefore so is u = «** (11.5.2). We remark that it is essential for the truth of (12.16.4), (12.16.5), (12.16.6), and (12.16.7) that the space E should be complete (cf. Problems 21 and 22). (12.16.8) (Banach's theorem) Let E, F be two Frechet spaces and let ubea continuous linear mapping ofE into F. Then either u(E) is meager in F, or else u(E) = F. In the latter case, if N is the kernel of u and E -» E/N -^ F the canonical factorization of u, the mapping v is an isomorphism of the Frtchet space E/N (12.14.9) onto F (in other words (12.12.7), u is a strict morphism of E onto F). Suppose that u(E) is not meager in F. By virtue of (12.13.1) and (12.12.7), the theorem will be proved if we can show that, for each neighborhood V of 0 in E, w(V) is a neighborhood of 0 in F. There are two steps to the proof. (12.16.8.1) Let E, F be two topological vector spaces, and u a linear mapping ofE into F such that u(E) is not meager in F. Then for each neighborhood V 0/0 in E, the closure w(V) of the image ofV is a neighborhood 0/0 in F. Let W be a balanced neighborhood of 0 in E such that W + W <= V ((12.13.1) and (12.8.3)). Then for each *eE there exists an integer n^ 1 such that x e «W. Consequently u(E) is the union of the sets w(wW) = n - w(W). Since u(E) is not meager, at least one of the sets n • w(W) has an interior point, and therefore so does #(W). But since -u(W) = w(W), we have ~w(W) = w(W). So if y0 is an interior point of w(W), then so is -j>0, and therefore 0=y0 + (-J0) is an interior point of w(W) 4- w(W) (12.8.2). But w(W) + w(W) is contained in the closure of w(W) + w(W) = w(W + W)c w(V), and so (12.16.8.1) is proved.ubspaces of E. Deduce that T(J? - x) contains only one -^-invariant point.